Small-Cancellation Disc Diagrams and the Torsion Toolkit: Examples

At each of $u,v$ the link is a path with three vertices (the three edge germs) and two edges (the two corners), so its Euler characteristic is $3-2=1$. Its angle sum is $1\mathrm{/}2+1\mathrm{/}2=1$, giving $k(u)=k(v)=2-1-1=0$ by [F1].

Each face is a bigon with two corners of angle $1\mathrm{/}2$, giving $k(f)=1-(2-2)=1$. Thus total curvature is $0+0+1+1=2$, agreeing with $2(2-3+2)=2$ in [F2]. These calculations include all four corners once in the face total and once in the vertex total with opposite sign.

By [F1] and the three-shell description [F2], the exterior length is $N-a-b-c$. The hypotheses give $a+b+c<3N\mathrm{/}6=N\mathrm{/}2$, hence $N-a-b-c>N\mathrm{/}2$. Equality is excluded because all three piece bounds are strict.

For the displayed instance, $3<19\mathrm{/}6$ because $18<19$, and $19-3-3-3=10>19\mathrm{/}2$. Thus the interior part has length nine and the exposed part length ten, verifying the asserted instance.

The symmetrised set is exactly $\{a^7,a^{-7}\}$. Its two words start in different letters, so have no nonempty common prefix; equal positive rotations all give the first word and equal inverse rotations the second. Thus there are no pieces and $C^{\mathrm{\prime }}(1\mathrm{/}6)$ holds vacuously by [F1]. The one-letter word $a$ cannot be a proper power of a nonempty shorter word, so it is a root by [F2].

Every one-generator word freely reduces to $a^j$ for an integer $j$, since any change of sign in the string creates an adjacent inverse pair. The relation $a^7=1$ reduces the exponent modulo seven, so every element is one of $1,a,\dots ,a^6$. The exponent sum modulo seven is unchanged by free cancellation and by inserting or deleting any conjugate of $a^{\pm 7}$: the conjugating exponents cancel and the relator contributes a multiple of seven. It therefore defines a homomorphism from the quotient to the additive residues modulo seven, taking $a^j$ to the residue of $j$.

The seven displayed elements have distinct images, so they are distinct; step 1.2 also proves that they exhaust the quotient. In particular $a^7=1$ and $a^j\mathrm{\ne }1$ for $1\le j\le 6$. This proves exact order seven, while keeping the literal word root $a$ distinct from its quotient image.

The single closed edge is finite, connected, planar and contractible, hence is a diagram by [F1]. Its face-label condition is vacuous. Its outer word from $u$ is $a{a}^{-1}$, a nonempty word freely reducing to the empty word and therefore null in the given free group. It is not freely reduced.

There are no faces, so there cannot be an exposed face or shell. Both endpoints have one edge germ and no corners, giving curvature one each by [F2], and the total is two. Thus positive curvature is entirely carried by spur tips, and the nonempty null boundary in step 1.1 refutes the assertion.